The distance function from a real algebraic variety

Giorgio Ottaviani; Luca Sodomaco

doi:10.1016/j.cagd.2020.101927

The distance function from a real algebraic variety

Giorgio Ottaviani, Luca Sodomaco^*

^*Corresponding author for this work

Department of Mathematics and Systems Analysis

University of Florence

Research output: Contribution to journal › Article › Scientific › peer-review

4 Citations (Scopus)

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Abstract

For any (real) algebraic variety X in a Euclidean space V endowed with a nondegenerate quadratic form q, we introduce a polynomial EDpoly_X,u(t²) which, for any u∈V, has among its roots the distance from u to X. The degree of EDpoly_X,u is the Euclidean Distance degree of X. We prove a duality property when X is a projective variety, namely EDpoly_X,u(t²)=EDpoly_X^∨,u(q(u)−t²) where X^∨ is the dual variety of X. When X is transversal to the isotropic quadric Q, we prove that the ED polynomial of X is monic and the zero locus of its lower term is X∪(X^∨∩Q)^∨.

Original language	English
Article number	101927
Journal	COMPUTER AIDED GEOMETRIC DESIGN
Volume	82
DOIs	https://doi.org/10.1016/j.cagd.2020.101927
Publication status	Published - Oct 2020
MoE publication type	A1 Journal article-refereed

Keywords

Euclidean distance
Euclidean Distance degree
Euclidean Distance polynomial
Isotropic quadric
Polar degrees
Real algebraic variety

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10.1016/j.cagd.2020.101927

SCI_Ottaviani_The_Distance_Function.1-s2.0-S016783962030114X-mainAccepted author manuscript, 1.01 MBLicence: CC BY-NC-ND

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AB - For any (real) algebraic variety X in a Euclidean space V endowed with a nondegenerate quadratic form q, we introduce a polynomial EDpolyX,u(t2) which, for any u∈V, has among its roots the distance from u to X. The degree of EDpolyX,u is the Euclidean Distance degree of X. We prove a duality property when X is a projective variety, namely EDpolyX,u(t2)=EDpolyX∨,u(q(u)−t2) where X∨ is the dual variety of X. When X is transversal to the isotropic quadric Q, we prove that the ED polynomial of X is monic and the zero locus of its lower term is X∪(X∨∩Q)∨.

KW - Euclidean distance

KW - Euclidean Distance degree

KW - Euclidean Distance polynomial

KW - Isotropic quadric

KW - Polar degrees

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The distance function from a real algebraic variety

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Keywords

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